Exercises for Section 10.1.  Basic Properties of Conformal Mappings

Exercise 1.  State where the following mappings are conformal.

1. (a)   .
Solution 1 (a).

1. (b)   .
Solution 1 (b).

1. (c)   .
Solution 1 (c).

1. (d)   .
Solution 1 (d).

1. (e)   .
Solution 1 (e).

1. (f)   .
Solution 1 (f).

Instructions For Exercises 2-5.
Find the angle of rotation    and the scale factor   of the mapping    at the indicated points.

Exercise 2.     at the points  .
Solution 2.

Exercise 3.     where    at the points  .

Remark.  .
Solution 3.

Exercise 4.    ,  where  ,  at the points  .

Remark.  .
Solution 4.

Exercise 5.    at the points  .
Solution 5.

Exercise 6.  Consider the mapping  .

If  ,  show that the lines    are mapped onto orthogonal parabolas.
Solution 6.

Exercise 7.  Consider the mapping  ,  where   denotes the principal branch of the square root function.

If  ,  show that the lines    are mapped onto orthogonal curves.
Solution 7.

Exercise 8.  Consider the mapping  .

Show that the lines    are mapped onto orthogonal curves.
Solution 8.

Exercise 9.  Consider the mapping  .

Show that the line segment  ,  and the vertical line  , where   are mapped onto orthogonal curves.
Solution 9.

Exercise 10.  Consider the mapping  ,  where    denotes the principal branch of the logarithm function.

Show that the positive -axis and the vertical line    are mapped onto orthogonal curves.
Solution 10.

Exercise 11.  (Indirectly Conformal Mapping)  Let     be analytic at    and  .
Show that the function    preserves the magnitude, but reverses the sense, of angles at  .

Solution 11.

Exercise 12.  If    is a mapping, where    is not analytic, then what behavior would one expect regarding the angles between curves?
Solution 12.

(c) 2008 John H. Mathews, Russell W. Howell